Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 24

Determine the derivative of the function $\displaystyle y = \frac{x}{\sqrt{7 - 3x}}$


$
\begin{equation}
\begin{aligned}
f'(x) &= \frac{d}{dx} \left( \frac{x}{\sqrt{7-3x}} \right)\\
\\
f'(x) &= \frac{(7-3x)^{\frac{1}{2}} \frac{d}{dx} (x) - (x) \frac{d}{dx} (7-3x)^{\frac{1}{2}}}{(\sqrt{7-3x})^2}\\
\\
f'(x) &= \frac{(7-3x)^{\frac{1}{2}}(1) - (x) \left( \frac{1}{2}\right) (7-3x)^{\frac{-1}{2}}\frac{d}{dx}(7-3x)}{7-3x}\\
\\
f'(x) &= \frac{(7-3x)^{\frac{1}{2}}- \left( \frac{x}{2}\right) (7-3x)^{\frac{-1}{2}} (-3) }{7-3x}\\
\\
f'(x) &= \frac{(7-3x)^{\frac{1}{2}}+ \frac{3x}{2(7-3x)^{\frac{1}{2}}}}{7-3x}\\
\\
f'(x) &= \frac{2(7-3x)+3x}{2(7-3x)^{\frac{1}{2}}(7-3x)}\\
\\
f'(x) &= \frac{14-6x+3x}{2(7-3x)^{\frac{3}{2}}}\\
\\
f'(x) &= \frac{14-3x}{2(7-3x)^{\frac{3}{2}}}
\end{aligned}
\end{equation}
$